Towards models of gravitational waveforms from generic binaries: A simple approximate mapping between precessing and nonprecessing inspiral signals

One of the greatest theoretical challenges in the buildup to the era of second-generation gravitational-wave detectors is the modeling of generic binary waveforms. We introduce an approximation that has the potential to significantly simplify this problem. We show that generic precessing-binary inspiral waveforms (covering a seven-dimensional space of intrinsic parameters) can be mapped to a two-dimensional space of nonprecessing binaries, characterized by the mass ratio and a single effective total spin. The mapping consists of a time-dependent rotation of the waveforms into the quadrupole-aligned frame and is extremely accurate (matches $>0.99$ with parameter biases in the total spin of $\Delta \chi \le 0.04$), even in the case of transitional precession. In addition, we demonstrate a simple method to construct hybrid post-Newtonian–numerical relativity precessing-binary waveforms in the quadrupole-aligned frame and provide evidence that our approximate mapping can be used all the way to the merger. Finally, based on these results, we outline a general proposal for the construction of generic waveform models, which will be the focus of future work.

Figure 1

The first panel shows the real part of the ($\ell =2$, $m=2$) mode with $\widehat{J}$ initially aligned with $\widehat{z}$, for the $1:10$ binary described in the text. The second panel shows the same quantity, but now with $\widehat{L}$ initial aligned with $\widehat{z}$. It is clear that if we are searching for signals with a monotonically increasing amplitude (as with nonprecessing binaries), we may easily class the signal in the lower panel as a glitch in the data.

Figure 2

The top panel shows the magnitude of the (2,2)-mode for the same strongly precessing case as in Fig. 1 over the whole length of the evolution, and over a length of the first 10000 M in the lower panel. The source frame was chosen such that ${\widehat{J}}_0\simeq (0,0,1)$, and $\overrightarrow{L}$ and ${\overrightarrow{\chi }}_2$ are initially orthogonal to each other with $\Vert {\overrightarrow{\chi }}_2\Vert =0.75$; the smaller black hole is not spinning. The close-up of the waveform magnitude over a shorter timescale reveals strong amplitude modulations.

Figure 3

The two panels show the angles found in an example of the maximization routine. The first panel shows the inclination angle ${\beta }_{\mathrm{MAX}}$ vs time, the second panel shows the azimuth ${\gamma }_{\mathrm{MAX}}$ vs time over the full length of the PN inspiral.

Figure 4

QA magnitude for the $q=10$ configuration considered in Figs. 1, 2. The top panel shows the complete waveform, while the lower panel zooms in on the first 10000 M. We see that the oscillations in the amplitude have been reduced and simplified from those in Fig. 2.

Figure 5

The absolute value of the GW strain for a precessing binary, as viewed at an arbitrary inclination of 2.8 rad from ${\widehat{J}}_0$. The signal includes all $\ell =2$-modes. The true signal (black) has the finer structure; the signal with the lower-amplitude high-frequency oscillations (red) was generated by twisting a nonspinning $q=3$ waveform with the inverse QA angles. The dotted line shows the amplitude of the original nonspinning waveform.

Figure 6

Evolution of $\overrightarrow{J}$ (red) and $\overrightarrow{L}$ (blue) plotted on the unit sphere, where $\overrightarrow{J}$ is initially aligned with the direction (0,0,1). The left panel shows the evolution of these two directions for a case of simple precession. The precession cone described by $\widehat{J}$ is very small in comparison to the one described by $\widehat{L}$, and appears on the figure as only a dot at the end of the vertical arrow. The right panel shows the same characteristic directions for a case of transitional precession. In this case $\overrightarrow{J}$ clearly moves along the unit sphere away from its initial direction (to the right side of the sphere) and separates from $\overrightarrow{L}$, which moves to the left side of the sphere in the figure.

Figure 7

The panel shows the magnitudes of the (2,2)-modes for the transitional precession case before (red; lower curve) and after (blue; upper curve) the quadrupole alignment was applied. The change of the direction of $\widehat{J}$ at $t=1.587\times {10}^6\mathrm{M}$ is indicated by the vertical line. A strong modulation is introduced into the original waveform at that time, which is completely removed after quadrupole alignment.

Figure 8

The two panels show the two Euler angles $\beta$ and $\gamma$ determined by the quadrupole-alignment procedure for the transitional case. The time when the $z$ component of $\overrightarrow{J}$ changes sign is indicated by the vertical line.

Figure 9

The PN (red, from $t=-300M$ to $t=100M$), NR (green, from $-100M$ to $300M$) and hybrid (dashed black) waveforms near the matching time ($t=0$). The PN and NR waveforms are blended together in the window $\Delta t=[-100,100]$, indicated by the shaded region.

Figure 10

Hybridization of the QA angles $\beta$ and $\gamma$. Upper two panels: the black (dotted) lines indicate the inspiral PN values, the red (dashed) lines indicate the later NR values, and the green (solid) lines indicate the hybrids. The lower figure shows the evolution of the aligned QA directions, where here the black line indicates long PN inspiral of duration $2.9\times {10}^5\mathrm{M}$, and the red line indicates the NR results up to merger.

Figure 11

Matches between QA and nonprecessing hybrids, for our standard $q=3$ configuration. The horizontal axis represents the frequency at which both waveforms are cut off in the match calculation, and indicates that the two hybrids agree well (match $>0.97$) right up to the merger, indicated by the vertical line.